MITx 6.86x – Machine Learning with Python-From Linear Models to Deep Learning

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MITx 6.86x – Machine Learning with Python-From Linear Models to Deep Learning

Course

/ Course Overview / Course Overview, Calendar, and Grading Policy

1. Course Introduction and Overview

Have you ever wondered how your cellphone can detect objects and discriminate between boats and a bridge? Or were you ever surprised when Amazon was able to recommend you a book that you didn’t even know that you wanted it. So those are just two examples of what machine learning technology can do for you today. But imagine what it can do in all other areas of our lives. And part of our job at MIT is actually to imagine this future. And here we’re working on all different application of machine learning and AI, ranging from drug discovery, identifying who may be sick, or developing new robotics applications, or helping people to live happier and longer. So there are lots and lots of applications out there, but what’s so interesting about it– that in terms of algorithms, there is a relatively small toolkit of algorithms you need to learn to understand how these different applications can work so nicely and be integrated in our daily life. So that’s what this class is about– understanding this technology. And I hope that, by the end of this class, you would actually know how your cellphone can detect bridges.

Hello. I’m Tommi Jaakola. I’m a professor here in computer science. And I’m going to be co-teaching this course with Professor Barzilay. I started my career as a theoretical physicist turned into a computational neuroscientist. And now I’m back in computer science. That path sort of reflects the diversity of tools, methods, concepts that are integral to machine learning. So let’s talk a little bit more about the structure of the course and the basic layout of machine learning problems. So Professor Barzilay already mentioned that you really need to learn only a few basic machine learning algorithms in order to be able to solve realistic interesting problems in practice. So let’s look at a little bit how these methods are laid out more broadly and how the course is structured accordingly. The biggest piece in terms of setting up tasks for machine learning methods it’s called supervised learning. And I will use a simple S here for supervised. It is when you’re given an example, such as an image, and a target, such as a label that it’s a dog. And your task as a machine learning algorithm is to figure out how to produce that label from that image. So in this case, the examples– and machine learning methods always learn from examples. So in this case, the example has the correct answer already given. So you’re learning essentially in their teacher-student setup. Those are supervised learning tasks. Now there is another piece here, which is called an unsupervised learning. And I will use US here as a place holder for that. Unsupervised learning is essentially a generalization of that, when you no longer are given the target exactly what you should do. So think about modeling a language. So the role of the machine learning method now is to try to generate natural language sentences. Or modeling images, its task is to learn based on just observing images, to reproduce such natural images, or molecules, or what have you. OK. So the major distinction between these two is that here what you need to produce is given explicitly as a target. Here, you need to figure out the mechanism of how to generate things. That’s unsupervised learning. Now, finally, both of these are essentially just making predictions, trying to create examples or predicting what the label might be for an image, what the image is about. The third part is actually acting in the world. And I will use RL for reinforcement learning for short. It is when you’re trying to figure out how to act optimally in the world. So it involves making predictions. So it involves aspects of these too, modeling the world. But also it has a goal. It is trying to achieve a certain objective. For example, a robot arm here trying to grasp an object, that can be formulated as a reinforcement learning task, where the method will try to figure out from the failure– it tries things. It may succeed or fail. And based on that, tries to figure out how to do it better. Now, the course here covers essentially the caricature versions of each of these pieces so that you learn the basic components. Lots of the interesting stuff here in terms of solving practical problems here happens in the mix of these categories. So the first three units in an increasing complexity try to figure out how to solve supervised learning tasks from a very simple linear classification method to more complex neural network algorithms. Here in an unsupervised learning, we talk about probabilistic models, how to generate examples, and look at things like mix and match. Finally, we will introduce you to basic reinforcement learning algorithms, what the problem is, and how to solve such problems. And overall, by learning all of these components, you will actually be able to mix and match these techniques to solve very interesting problems in practice.

Course / Unit 0. Brief Prerequisite Reviews, Homework 0, and Project 0 / Brief Review of Vectors, Planes, and Optimization

1. Brief Review of Vectors, Planes, and Optimization

Hi. In this short video, we’re going to talk about points and vectors and how they relate to machine learning. So first, the machine learning, there are two things. There are points– data points which you have in your data sets, which you’ll be working with. And then you also want to be able to describe them as how they relate in n-dimensional space. So let’s start off with getting a distinction between the two. So if we draw a three-dimensional coordinate system, like so, we’ll have a point here. We’ll call it x. It might be 3, 2, 2. And this defines a point in our three-dimensional space. We can write three-dimensional space as this x lives in this three-dimensional space. We can also write this as a vector starting from the origin, where we start over here. And we view it in terms of a displacement from 0 comma 0 comma 0. So we have both the idea of this vector, so this vector is x, which goes from 0 to this point. And we can also view it as just the point in space. In this course, we’ll be viewing this description of a point and vector pretty much interchangeably. So let’s write this vector as you’ll commonly seen this course. So x here, we denote as all of its coordinates. So we can either write it like this, as a point, or like this, as a vector. And so it’s important to note that if we index this by dimension, dimension xi is going to be the i-th coordinate in this vector. So let’s say that xi is equal to x comma 2. And so this will be 1, 2. So it’s also important to be able to understand how vectors can relate to each other. So, in particular, one thing you might be interested in is seeing how, if you add vector [INAUDIBLE],, what happens. So if we go back to our coordinate system, we can see that vector addition works like this. Let’s say we have a vector here, which we can call A. And then we have a vector here, which we call B. And so one thing we might want to see here is, actually, what is this vector between the two? So here, we can just do standard vector addition. We know that A plus this vector C is equal to B. So C is equal to B minus A. Another question we might want to ask about vectors is– how big are they? How long are they? How can I understand what is the size of this vector? So one important thing, which we’ll be using in this class is the conception of a norm. So if you recall from Euclidean distance, the norm of this vector in this space is going to be equal to– we denote it like this– the square root of all of its items squared. So this will be 3 squared plus 2 squared plus 2 squared. In general, this will just be denoted as the sum from i equals 1 to n of our n-dimensional vector of each element xi squared. And typically, we’ll take the square root of that. Another important concept is the concept of a dot product. This has a relationship as to how vectors are arranged relative to each other. So there are two definitions for a dot product. We denote it as x dot y. And let’s start first with the algebraic definition, which is just the sum from i equals 1 to n of the components times each other– so xi times yi. This is the algebraic definition. So, for example, if we had this vector x, which is 3, 2, 2, and we had another vector– let’s say– let’s call this y over here. We’ll just write it over her for now. So y is equal to 1, 1, 1. All right x dot y is equal to the square root of 3 times 1 plus– sorry, this is not the square root. Ah, I messed it up. So x dot y is equal to 3 times 1 plus 2 times 1 plus 2 times 1, which is equal to 3 plus 2 plus 2, which is equal to 7. There is also a geometric definition, where x dot y is equal to the magnitude of x, the norm, and y times the cosine of the angle between them. So, in this case, if we have two vectors here, this is x, and this is y. We denote this as theta, which is the angle between them. And we have this relationship for the dot product. And so the two definitions are equivalent to each other. And you can be able to use them to do several interesting things, which you’ll see in homework 0.

Hi. In this short video, we will be talking about planes and other low-dimensional subspaces of higher n-dimensional space. In this class, we will commonly be working with n-dimensional spaces, where points are described in n dimensions. For example, we can have a vector, x, which is equal to some, let’s say, three-dimensional space– x1, x2, x3. And we denote this as living in r3. So oftentimes, we will be interested in vectors which live in a separate subspace of this higher space. So to explain what I mean, let me draw a picture. So here is r3. Our vector here might live over here. This is our vector. But we are actually going to be interested in those points which are described by some linear relationship. For example, we could have a plane, which is all the points which fit on this slice. In two dimensions– color this in. In two dimensions, we might have a line. Here, this could be described by, say, the plane z equals 3. And x and y are free to be whatever. And in this case, this would be described by line y equals 1, for example. So this becomes particularly relevant in machine learning when we want to be able to describe the way that points are arranged in this space and what side of the space they fall on. For example, let’s look at a simple example where we’re trying to draw classification boundaries. So in this sense, we can have some points which are x’s and some points which are o’s. And we want to ask ourselves, what is the separating hyperplane? By hyperplane, I mean a line here, as in n minus 1 dimensions, that separates these two sets of points. So here we might find a line, here, which separates our space. And so on either side of this plane, a line in two dimensions will follow our data points. So here we can say, if everything less than this plane, on the other side of this plane, these are all o’s. And on the other side, they’re all x’s. And that’s one of the decision boundaries that we’ll be trying to find in machine learning. So how do we define this line? And here we had just wrote this out as z equals 3 and y equals 1. But there’s actually a particular definition in which you can use to define what a plane is. And particularly, you want to define a linear relationship between two dimensions that satisfy all of these points being on this surface. So let’s say that we have a plane here. So let’s say, [? “defining ?] a [? plane.” ?] If we have a plane in, say, three dimensions– let’s draw it over here. We define this plane as having a normal, as in there is some vector here, which we are taking to be as perpendicular to all of the other points which fall on this plane. Let’s call the vector of this normal theta. [][Additionally, this plane has an offset from the origin.] So it’s over here– o prime. This is o. And this is its offset. Let’s call this x hat. So now we want to find all of the lines, all of the points which fall on this plane. So if we take another point, x, it will only be on this plane if its vector is perpendicular– draw it over here– if its vector is perpendicular to theta. So we can write it like this. So this vector here is going to be x minus x hat dot theta is equal to 0. So x dot theta minus x hat dot theta is equal to 0. This is always going to be a constant. So we can just define our plane as all of the points which satisfy x dot theta plus some offset– theta 0 is equal to 0– where these two are defined to be equivalently. And this is the definition of our plane. So any point which satisfies this relationship, this linear relationship, lies on this plane. End of transcript. Skip to the start.

Hi. In this video, I’m going to talk about the loss function, gradient decent algorithm, and some tools on how to compute the gradients. The basic tool us we will use very often is the chain rule. Now the first two topics will be covered later on in the lectures, and here I am just giving you some basic idea, and some intuition about the things that we’re going to talk about. OK so loss function. The loss function is some way for us to value how far is our model from the data that we have. OK, so let’s look at some simple linear regression model, which is just a model that we can look, if we have just one-dimensional data, like this. OK we have some input value x, and we use some parameters theta 1 and theta 2, which we want to learn in order to predict some y. Let’s say for example, if we want to have a use case that x is the height of some person, and y is shoe size. OK, so let’s say that we start with the model, and now we have some data points that are something like that. OK, obviously our model can be improved, and we want to learn the better parameters for theta 1 and theta 2 to better described our data. So how can we do that? Let’s first define the error. Now the error can be defined many ways. Let’s for example say that we take it as the vertical distance between our prediction to the real value. OK, so if we write this as an equation, our loss function based on x and y will be just the vertical distance between the two points. OK, this is for this is for one example from the data, but we can actually take a set off examples, and then take the mean of them. And this is actually is this, right? Now we will learn about the actual loss functions that are being used. And don’t worry about that. We’ll see many more. But now, let’s, for example, say that we have some kind of a loss function that looks like this, where here I mean that this is the theta, and this is the value of the loss function. Let’s say that this is theta 1. So given this loss function, now, for specific parameters that they have, I can say that right now I’m on this place on the graph. And what I actually want to do is to minimize the loss function, right? If the loss function is minimized, it means my model best describes the data. OK now in simple cases, I might just solve it directly and find the minimum of the loss function. But many times, our models would be very complex, and we’ll have a lot of data. And actually what we want to do is to use some iterative algorithm. The most common algorithm to use is the gradient descent. The gradient descent is actually very simple concept. What we will do is we will take the loss function that we get from our current set of data points. We will compute the gradient. The gradient is just the derivative of the function in this point. And we want to move against the direction of the gradient. OK, so if it’s here we want to move a bit here, and we’ll get to this point. So if we write it down, the update steps of the gradient descent, we will want to find a new value for data, which is the old value minus some gamma times the gradient of the loss function. OK and this gamma is what is called the learning rate. If it’s very high, then we might move like this and diverge. And if it’s very small, we might converge very, very slowly, or if our loss is actually– if we have some local minimums, we might be more prone to get stuck in them. This is about– the loss function sometimes has been called the cost function or the race function. It is all the same. Now let’s talk a bit about how to compute gradients. So our models will many times be pretty complex functions. Let’s take, for example, let’s say that our prediction model is something like that. e to the minus theta x plus theta 2. So now we want to derive that. Let’s say in terms of– sorry, in terms of, theta 1. OK so what the chain rule tells us, let’s write it down here. But if I have some function that I want to derive according to x, I can actually derive it like that in part. OK, this is all. So if I want to derive this function now, I can actually write something like this. OK, and what is this z? So I will define z just to be this part of the equation. OK and now actually I have this linear formula, and on top of that I have this more complex function, which is actually called the sigmoid function, and we will see it later on in the class. So I can actually look at my model as if I’m getting some input x, and then I’m applying some linear transformation on it, which gives me z. And then I’m applying that sigmoid function to get my prediction. So now all we need to do is to solve those two derivatives. OK now, obviously, if we– let’s first do actually the z. Let’s start with a simple one. dz to d theta 1, which will be only x. And now I need to compute dy to dz. So actually what I want to compute now is this guy. OK, and this is actually, as I said, is a sigmoid function, and you can compute it yourself, but what you will get is that the derivative is this minus z squared. OK, so now I can just put those things together, and use the chain rule to get my final derivative. Just x times that sigmoid derivative squared. OK, now I can plug in back z to get my derivative. If I want to do the same thing for theta 2, then it’s simply this. 1 plus, yep. OK, so this is the chain rule, and this is a very useful tool to derive eyes more complex functions, and we will surely need to use it for deep learning later. And that’s it.

Course / Unit 0. Brief Prerequisite Reviews, Homework 0, and Project 0 / Homework 0

1. Objective

2. Sums and Products

3. Function Properties

4. Points and Vectors

5. Planes